Question

show that any one of the number n +2 ,n and n+ 4 is divisible by 3

Answer 1

Let n be a positive integer in the form 3q, 3q + 1 or 3q + 2

Case 1 when n = 3q
n = 3q
, this is divisible by 3

n +2 = 3q + 2, 
this is not divisible by 3

n + 4 = 3q + 4, this is also not divisible by 3
Case 2
when n = 3q + 2
n = 3q + 2, this is not divisible by 3

n + 2 = 3q +2+2=3q+4, this is not divisible by 3

n+4 = 3q+2+4 = 3q +6 = 3(q+2), this is divisible by 3

Case 3 

when n = 3q + 1

n= 3q + 1, not divisible

n+2 = 3q+1 + 2= 3q + 3 = 3(q+1)
divisible

n + 4 = 3q + 1 + 4 = 3q + 5,
not divisible.

Answer 2

This is impossible . We can give simple reasons .The n is the number which is divisible by 3. But 2 and 4 is not divisible which is added to the divisible number and thus it becomes indivisible for both +2 and+4.e.g -3+2= 5 which matches all the numbersmatches all the numbers.